What Are the Properties of the Commutator in the Dilation Operator?

In summary, the Dilation operator is a mathematical operator in quantum mechanics that describes the change in state of a quantum system over time. It affects the wave function of a system by expanding or contracting it. It is closely related to the Hamiltonian operator and is used to calculate the probability of a system being in a certain state at a specific time. It cannot predict the exact state of a system, but is used in practical applications such as calculations and simulations in the development of quantum technologies.
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Homework Statement



Concicer the dilation operator

[itex] D = \vec{r} * \vec{p}[/itex]

Compute [itex] [D,\vec{r} ] and [D, \vec{p}] [/itex]

Homework Equations



[itex] p = - i * hbar [/itex]


The Attempt at a Solution


I think the question is really if [itex] [D, \vec{r}] [/itex] commutes

I got this:
[itex] D = \vec{r} * \vec{p}[/itex]
[itex] D = x p_x + y p_y + z p_z[/itex]


for the x direction:

[itex][D, r] = [D, x] = [x, x p_x][/itex]

where do i take it from here?
 
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  • #2
You have absolutely no idea ?? Pick up your book and find those brackets (called <commutator>) and the properties they have.
 

1. What is the Dilation operator in quantum mechanics?

The Dilation operator, also known as the time-evolution operator, is a mathematical operator used in quantum mechanics to describe the change in state of a quantum system over time. It is used to calculate the probability of a system transitioning from one state to another at a certain point in time.

2. How does the Dilation operator affect the wave function of a quantum system?

The Dilation operator acts on the wave function of a quantum system to evolve it from one state to another. It can cause the wave function to expand or contract, depending on the specific properties of the system and the time interval being considered.

3. What is the relationship between the Dilation operator and the Hamiltonian operator?

The Dilation operator is closely related to the Hamiltonian operator, which represents the total energy of a quantum system. The Dilation operator is derived from the Hamiltonian and is used to calculate how the wave function of a system changes over time.

4. Can the Dilation operator be used to predict the exact state of a quantum system at a future time?

No, the Dilation operator can only provide the probability of a system being in a certain state at a specific time. It does not give a definite prediction of the exact state of the system, as quantum mechanics is inherently probabilistic.

5. How is the Dilation operator used in practical applications of quantum mechanics?

The Dilation operator is an essential tool in the field of quantum mechanics. It is used in various calculations and simulations to analyze the behavior of quantum systems and make predictions about their future states. It is also used in the development of quantum technologies, such as quantum computers and quantum cryptography.

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